Portfolio Optimization

Transcription

1 Portfolio Optimization The Martingale Approach Master Thesis Patrick Deuß Supervisor University of Wuppertal Prof. Dr. Michael Günther University of Wuppertal Faculty of Mathematics and Science May 26

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3 To my family. I thank Alexandra Claßen for her love, patience and understanding. Special thanks go to my supervisor Prof. Dr. Michael Günther, Prof. Dr. Manfred Mendel, Christian Kahl and Florian Unkel for their support and the time they spent for me. Patrick Deuß, May 26

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5 Contents List of figures List of tables Abbreviations Introduction iii iii v vii 1 Stochastic analysis Stochastic processes and martingales The Ito calculus Solutions of stochastic differential equations A model of the financial market Modelling prices Portfolio processes The complete market Assumptions for the complete market Portfolio optimization The continuous-time portfolio problem The martingale approach Optimal portfolios Comparison of coefficients A solution for the portfolio problem Applications and numerical tests Logarithmic utility Data and implementation A comparison Conclusion 73 A Mathematical tools 75 A.1 Probability spaces A.2 Stochastic tools and probability beliefs A.3 Properties of utility functions A.4 KKT-conditions and Lagrangian multipliers A.5 The DAX corporations B Proofs 83 i

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7 List of Figures 1 Efficient portfolios according to Markowitz viii 1.1 Brownian motions and their expectations A left-continuous step function φ 4 t with φ i = ϕ i, i = 1,..., Possible representations for 1 2 B2 t (ω) t 2 and B t(ω) Possible stock prices and the stock price expectation The discount factor δ(t, ω) Utility function U t (x) = exp(.5t) x Extended utility function U t (x) = exp( t) 1 2 (x 1 2 ) Efficient stocks and a variation of the CML for the DAX List of Tables A.1 The 3 DAX corporations, April 28, A.2 Market capitalization in millions of EURO, May 5, iii

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9 Abbreviations Please note that all vectors in this thesis are defined as column vectors. 1. a b := min{a, b} 2. R + := {x R x > } 3. R + := {x R x } 1 4. e n :=. R n 1 5. w.l.o.g. := without loss of generality 6. P -a.s. := P almost surely 7. w.r.t. := with respect to 8. P -a.e. := P almost every/everywhere 9. f + := max{, f}, f := min{, f} 1. B n := Borel-algebra on R n 11. B[, t] := Borel-algebra on the interval [, t] 12. B := Borel-algebra on R 13. a t := the transposed of vector a x x 1m 14. x R n m :=..... x n1... x nm with x ij R, i = 1,..., n and j = 1,..., m 15. f C (, ) := f is continuous on the interval (, ) 16. f C n, n N := f is n-times continuously differentiable 17. f C n,m, n, m N := f is n-times continuously differentiable in the first variable and m-times continuously differentiable in the second one 18. s.t. := such that 19. a b := a or b 2. lim f(x) := lim f(a + 1 x a n N n ) as right-hand limit 21. p.e. := for example v

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11 Introduction How can we optimize our portfolio? For sure, there are many ways to find solutions for this question. One mathematical attempt is presented in this thesis: portfolio optimization via the martingale approach. What do we understand by a portfolio? Normally, we declare a portfolio as a compound of all conceivable instruments in the financial market (bonds, stocks, options, funds, all kinds of derivates, real estates, certificates, commodities etc. - the list could be amplified arbitrarily). To simplify the calculus and formula in this thesis we confine our portfolio to stocks and bonds. Please note that almost all other kinds of securities are either similar to stocks or can be replicated by them in an analytical-mathematical way (compare the strategy of duplication - option pricing). Portfolio optimization and its history The first one who made investment behavior more comprehensive in an analytical way was Harry Markowitz with his article Portfolio Selection in the Journal Of Finance, 1952 (Markowitz [1]). This ground-breaking work was the premier that showed that risk / volatility and expectation of securities can be understood as statistical measures, namely as variance and as expected value and it strongly influenced the portfolio management (the capital asset pricing model=capm is based on Markowitz explanation report) and portfolio optimization. Markowitz indicates how to compute efficient portfolios. In a mathematical sense, this computation is a calculation of optimal surfaces by linear or nonlinear programming. An efficient portfolio is one with minimum risk for given expectation or one with maximum expectation and given risk. Markowitz model is a discretetime one. The subsequent approaches for the portfolio problem, whether in a discrete-time or in a continuous-time market model, use deeper mathematics such as the stochastic control approach introduced by Merton [12, 13] in 1969 or the martingale approach in different versions (198ies), for example. The problem An investor is endowed with a certain starting capital which he has to allocate on his portfolio, the 3 DAX corporations and one riskless bond, for instance. During his investment (a finite time horizon) he can consume and redistribute the shares of the securities. This means we search for a continuous-time solution. His aim is to maximize the utility of consumption and the terminal wealth of his portfolio. How the investor chose the utility depends on himself. Furthermore, we suppose that the necessary data (expected returns, volatilities and correlations etc.) are given by the former performances and observations and the near future expectations of the assets and are available for the investor. vii

12 efficient portfolios expectation attainable portfolios risk / volatility Figure 1: Efficient portfolios according to Markowitz One solution This thesis presents one solution of the portfolio problem: the martingale approach. In our methodology we will follow predominantly Korn [2, 4]. We assume that our market is complete in a mathematical and economical sense (this will be explained later, see chapter 2) which is evident for this approach. Moreover, the solution requires a continuous-time market model. Concerning the composition of this thesis it must be pointed out that the reader is completely introduced to the subject of portfolio optimization: special previous knowledge is not needed with the exception of mathematical basics. He might find some helpful comments in appendix A, long proofs are attached in appendix B. Please note the abbreviations at the very beginning. In chapter 1 basic terms and tools of stochastic analysis such as stochastic differential equations or the Itô calculus are presented and provide a basis for chapter 2 and 3. The second chapter deals with the modelling of the complete financial market. Here, stock prices are created, for example. In chapter 3 the derivation of the martingale approach and a general solution of the portfolio problem are described. We apply the obtained solutions to a portfolio of the 3 DAX corporations and one bond (the initial problem) in chapter 4 and evaluate this numerical example. A conclusion is given in chapter 5. viii

13 1 1 Stochastic analysis In this chapter the mathematical basis will be created to handle the problems we will be faced with during this thesis. The composition should be selfexplanatory. In addition to that the reader might find some helpful annotations and explications in the appendix. 1.1 Stochastic processes and martingales If we want to model a financial market, we have to think about its basic instruments and properties. First, we need a sort of time structure connected with a description of the white-noise-effect or chance / random in the market. Assume that (Ω, F, P ) is a complete probability space. 1.1 Definition Let {F t } t I be a family of sub-σ-algebras of F and I R an interval with I := [, T ], T R +, (T = possible: I = [, )). If F s F t is held for s t, s, t I, then {F t } t I is a filtration. 1.2 Definition A set {(X t, F t )} t I such that i) {F t } t I is a filtration ii) {X t } t I is a family of random variables assuming values in R in which X t is F t -measurable with X : [, T ] Ω R (t, ω) X t (ω) := X(t, ω) is called stochastic process with filtration {F t } t I. We remark that i) for fixed t I, ω Ω, X t (ω) with ω X t (ω) is a random variable, ii) for fixed ω Ω, t I, X t (ω) with t X t (ω) is a function depending only on time t. We call this path or realisation of a stochastic process. iii) Abbreviation: X t := X(t, ω) := {(X t, F t )} t I Explanation for definitions 1.1 and 1.2 The σ-algebra F t, t I, will model the observable occurrences of a random variable X t till time t. If X t is F t -measurable, we can determine the values of X t at time t. Therefore, a filtration {F t } t I reflects a time structure. A stochastic process will be an elementary instrument to formalize random depending on time and the market situation.

14 2 Stochastic analysis As we will see later, one example of a stochastic process plays an important role in modelling stock prices: the Brownian motion. A one-dimensional Brownian motion B t (ω) is a stochastic process {B t } t I with the following properties: i) B t (ω) R t I, ω Ω, B : [, T ] Ω R ii) B t ( ) C ω Ω (i.e. continuous paths for fixed ω Ω) and a) B (ω) = P -a.s. b) B t (ω) B s (ω) N(, t s) for s t (stationary increases) c) B t (ω) B s (ω) is independent from B u (ω) B r (ω) for r u s t (independent increases) For the n-dimensional case we have B t (ω) = (B 1 (t, ω),..., B n (t, ω)) t such that B i (t, ω) are independent, identically distributed one-dimensional Brownian motions. The filtration for the Brownian motion can be defined on the one hand as F B t := σ{b s s < t} (natural filtration) or, on the other hand as F B t { } := σ Ft B N N F, P (N) = (Brownian filtration). Because of some technical reasons we will use the Brownian filtration. To prove that the Brownian motion exists as a stochastic process is very technical. For further information see Billingsley [15]. We note that the Brownian filtration {Ft B } t I is right- and left-continuous (see Karatzas and Shreve [16]). right-continuous : F t = F t + := left-continuous : F t = F t := σ F t+ɛ ɛ> ( ) F s s t 1.3 Definition A filtration {F t } t I fulfils the usual conditions, if i) {F t } t I is right-continuous and ii) F contains all sets N Ω with P -outer measure zero (P (N) = ). Explanation In the following we will use filtrations holding definition 1.3 which guarantees that our time structure is adequately smooth.

15 1.1 Stochastic processes and martingales 3 How can we describe and analyze a random variable under changed conditions? Which values or expectation does it assume if we underly a different σ-algebra for example? The answer to this question is given in the next definition (compare also Øksendal [5]) 1.4 Definition Let X : Ω R be a random variable with E[ X ] < and H F a σ-algebra. The conditional expectation E[X H] is the P -a.s. unique function such that i) E[X H] is H-measurable and ii) H H and random variables Z, which are H-measurable E[X H]dP = XdP Z E[X H]dP = Z XdP is held. H H Please note that uniqueness and existence of E[X H] are deduced from the theorem of Radon-Nikodym (see Bauer [6] or Michel [1]). For the calculus with E[X H] we take a closer look at the next theorem. 1.5 Theorem Properties of the conditional expectation E[X H] are i) E[E[X H]] = E[X] ii) E[X H] = X if X is H-measurable iii) E[X H] = E[X] if X is independent of H. Without proof. Ω Ω We will learn that one class of stochastic processes are well suited for the mathematical depiction of the financial market: martingales. 1.6 Definition Let {(X t, F t )} t I be a stochastic process with E[ X t ] < t I. If E[X t F s ] = X s P -a.s. s, t I, s t, {(X t, F t )} t I is called martingale. The process X t with a) E[X t F s ] X s is a supermartingale, b) E[X t F s ] X s is a submartingale. Explanation Let X n, n N, be the amount of money an investor has after n trading days on the market. Then a fair market would fulfil the martingale condition E[X n+1 F n ] = X n P -a.s.. That means that the expected amount of money after the (n + 1)-th participation on the market is the same as after n-days. A favorable market for the investor would be a submartingale (E[X n+1 F n ] X n ), an inconvenient market a supermartingale (E[X n+1 F n ] X n ).

16 4 Stochastic analysis 1.7 Corollary The one-dimensional Brownian motion B t is a martingale. Proof E[B t F s ] = E[B t B s + B s F s ] = E[B t B s F s ] + E[B s F s ] ( B s is F s -measurable) = E[B t B s F s ] + B s ( B t B s are independent of F s ) = E[B t B s ] + B s ( B t B s N(, t s)) = B s 1.8 Example Let X i (t) = µ i t + σ B t, µ i, σ R, t, i = 1, 2, 3, be a Brownian motion with drift µ and volatility σ. Then we have i) µ 1 > X 1 (t) is a submartingale, ii) µ 2 = X 2 (t) is a martingale, iii) µ 3 < X 3 (t) is a supermartingale. In the next figure the different drifts are defined by µ 1 = 1, µ 2 =, µ 3 = 1 whereas the volatilities are equal for X i (t), i = 1, 2, 3, that is to say σ = 1. 1 X 1 (t)=t+b t X i (t) and E[X i (t)]; i=1,2,3 5 5 E[X 3 (t)] E[X 1 (t)] E[X 2 (t)] X 2 (t)=b t 1 X 3 (t)= t+b t t Figure 1.1: Brownian motions and their expectations

17 1.2 The Ito calculus The Ito calculus Later, when we will model stock prices, the question will occur how to solve a stochastic integral such as X s (ω)db s (ω) (1.1) in which B t (ω) is a Brownian motion and X t (ω) a measurable, non-negative function with values in R, X t (ω) : [, T ] Ω R +. We assume that F t is a Brownian filtration fulfilling the usual conditions. Still (Ω, F, P ) is a complete probability space. The Ito integral To make a long story short, we have to realize that we cannot compute (1.1) in the Riemann-Stieltjes sense: If µ is a distribution for Y, µ C 1, with density ν(s) = dµ(s) ds, we solve the Riemann integral Y (s)dµ(s) = Y (s)ν(s)ds. This approach is not possible for (1.1) because the realisations of a Brownian motion B t (ω) are P -almost nowhere differentiable. Furthermore, we cannot define (1.1) as a Lebesgue-Stieltjes integral; the paths of B t have P -a.e. infinite variations. Hence, we need some sort of approximation for (1.1). The idea is the following: we will define a stochastic integral for elementary processes. Then we will show that stochastic processes (with certain restrictions) can be approximated by these elementary processes and we will finally find a solution for (1.1). 1.9 Definition A stochastic process φ (n) t (ω) is an elementary process, if i) t i R, i =, 1,..., n, n N, with = t < t 1 <... < t n = T ii) bounded random variables ϕ i (ω), i =,..., n, ϕ i <, ϕ : Ω R with such that a) ϕ is F -measurable b) ϕ i is F ti 1 -measurable φ (n) t (ω) = ϕ (ω)χ (t) + in which χ (ti 1,t i](t) := n ϕ i (ω)χ (ti 1,t i ](t) i=1 { 1 if t (ti 1, t i ] otherwise ω Ω

18 6 Stochastic analysis Note that χ is the indicator function, φ (n) t (ω) is F ti 1 -measurable for t (t i 1, t i ], the more n N grows, the smaller the intervals (t i 1, t i ] get. This means that with n the interval I = [, T ] is divided into infinite parts, so to say: (t i 1, t i ] t as n. Notation S := {X t X t is an elementary process}. 1.1 Example A path of an elementary process φ (n) t is a left-continuous step function. Figure 1.2: A left-continuous step function φ 4 t with φ i = ϕ i, i = 1,..., 4 Now, we define (1.1) for elementary processes Definition Let φ (n) t S, t I. The stochastic integral I [φ] is defined for t I as I t [φ (n)] (ω) := φ (n) s (ω)db s (ω) := n i=1 ϕ i (ω) [ B ti t(ω) B ti 1 t(ω) ] }{{} :=

19 1.2 The Ito calculus 7 Interpretation and remarks 1) For t (t i 1, t i ] it is φ (n) t ϕ i, ω Ω. 2) is the increase of the Brownian motion B t on (t i 1, t i ]. 3) We multiply the increases ( ) by the values of φ (n) t ( ϕ i ). 4) For φ t, θ t S, a, b R we have I t [aφ t + bθ t ](ω) = ai t [φ t ](ω) + bi t [θ t ](ω) 5) r φ s (ω)db s (ω) = r φ s (ω)db s (ω) φ s (ω)db s (ω) For the approximation of elementary processes to stochastic processes in general we need a new term concerning the measurability of processes Definition Let X t be a stochastic process. Then X t is F t -adapted if X : [, t] Ω R n is B[, t] F t B n measurable t I. (s, ω) X s (ω) Remember that {F t } t I is an increasing family of σ-algebras in Ω. Notation Let L 2 [, T ] := L 2 ([, T ], Ω, F, {F t } t I, P ) be the class of functions of X t (ω) = X(t, ω) : [, T ] Ω R such that i) X : (t, ω) X(t, ω) is B F-measurable ii) X(t, ω) is F t -adapted and [ ] T iii) E Xt 2 (ω)dt < (i.e. is bounded). Note that L 2 [, T ] is a vector space and that {(X t, F t )} t I is a R-valued stochastic process. Now, we want to approximate X L 2 [, T ] by φ (n) S, n N, in other words: lim n N φ(n) X With this observation we will show that I[X] = lim n N I [ φ (n)] is held thereby resolving our problem for (1.1) for X L 2 [, T ]. First, we introduce an important property of φ S.

20 8 Stochastic analysis 1.13 Lemma (The Itô isometry) For φ S, φ bounded, it is E φ (p) (s, ω)db s (ω) 2 = E φ (p) (s, ω) 2 dt t [, T ] (1.2) Proof: see appendix B. Use the definition of φ and compute E [ I t (φ) 2] with help of a case differentiation (i j and i = j). The result of the Itô isometry is of course very useful, because we do not need to integrate in dependence of the Brownian motion B t (ω). Instead, we[ calculate the ( ) ] 2 integral in dependence of time t. Vice versa we know then that E XdBs exists. With the next fundamental theorem we will show that an approximation as we mentioned above can be found Theorem Let X L 2 [, T ]. Then a sequence {φ (n) } n N, φ (n) S n N such that lim E n N ( X t φ (n) t ) 2 dt = (1.3) Proof: see appendix B. Write X t as elementary process φ n t with a suited indicator function. Show that this process converges to X t as n with help of theorem of dominated convergence A.4, appendix A.2. [ 1.15 Definition (The Itô integral) T ( Let X L 2 [, T ], φ (n) S, n N, with E n. Then I t [X](ω) is defined as I t [X](ω) := as limit in L 2 [, T ]. X s (ω)db s (ω) := lim n N X t (ω) φ (n) t ] ) 2 (ω) ds as φ (n) (s, ω)db s (ω) (1.4)

21 1.2 The Ito calculus 9 Remark i) I t [X](ω) exists (i.e. the limit (1.4) exists) and is independent of the choice of {φ (n) } n N as long as (1.3) is held. ii) φ (n) (s, ω)db s (ω) = n ϕ i (ω) [ B ti t(ω) B ] ti 1 t(ω) i=1 I t [X](ω) = lim n N i=1 n ϕ i (ω) [ B ti t(ω) B ti 1 t(ω) ] iii) The sequence of elementary processes {φ (n) } n N exists, proven by theorem (1.14) Corollary (The Itô isometry for functions in L 2 [, T ]) Let X L 2 [, T ]. Then is valid. E X t (ω)db t (ω) Proof Follows from equations (1.2) and (1.4). 2 = E X t (ω) 2 dt [ ] 1.17 Corollary T ( ) 2 Let X t L 2 [, T ], φ (n) t S, n N, with E φ (n) t X t dt as n. lim φ (n) t dt = X t dt in L 2 [, T ] Without proof. n N To illustrate the calculus with stochastic or Itô integrals, we regard the next example Example B s (ω)db s (ω) = 1 2 B2 T (ω) T 2 1dB s (ω) = B T

22 1 Stochastic analysis 3 2.5*B t 2.5*T and Bt B t.5*b t.5*t t Figure 1.3: Possible representations for 1 2 B2 t (ω) t 2 and B t(ω) Proof φ (n) s := n B tk χ (tk 1,t k ](s) lim φ (n) s = B s n N k=1 E n =E ( φ (n) s k k=1t k 1 ) 2 B s ds ( φ (n) s ) 2 B s ds ( φ (n) s B tk for s (t k 1, t k ]) n = k k=1t k 1 [ E (Btk B s ) 2] ds = n k k=1t k 1 V ar(b tk B s )ds n k n = (t k s)ds = [t k s 12 ] tk s2 k=1t k=1 t k 1 k 1 = 1 n (t k t k 1 ) 2 (as n, because t k t k 1 ) 2 k=1 This means as n (compare corollary 1.16 and definition 1.15) E (φ (n) s 2 T B s )db s φ (n) s db s B s db s

23 1.2 The Ito calculus 11 Furthermore, we have n B tk (B tk B tk 1 ) = k=1 = 1 2 B2 T = 1 2 B2 T = 1 2 B2 T n Bt 2 k k=1 n Bt 2 k + 1 n 1 Bt 2 2 k k=1 n Bt 2 k k=1 k=1 n B tk B tk 1 ( B tn = B T ) k=1 n B tk B tk 1 ( B t = B = ) k=1 n Bt 2 k 1 k=1 n ( 2 Btk B tk 1) k=1 With this we reach for lim n N = lim n N k=1 = 1 2 B2 T lim n B tk B tk 1 k=1 φ (n) s db s ( definition 1.11) n B tk (B tk B tk 1 ) n N k=1 n (B tk B tk 1 ) 2 } {{ } :=η φ (n) s db s = 1 2 B2 T T (η T in L 2 [, T ], without proof) And, for the other equation: n n n 1dB s = (B tk B tk 1 ) = B tk B tk 1 = B tn = B T k=1 k=1 k=1 For the generalization from a one-dimensional Itô integral to a multi-dimensional one we have: Notation X 11 (t, ω)... X 1m (t, ω) Let X(t, ω) :=....., {(X t, F t )} t I be a X n1 (t, ω)... X nm (t, ω) F t -adapted process with X ij L 2 [, T ] assuming values in R n m and filtration {F t } t I, {(B(t, ω), F t )} t I a m-dimensional Brownian motion with B(t, ω) = (B 1 (t, ω),..., B m (t, ω)) t.

24 12 Stochastic analysis Then we define X(s, ω)db s (ω) := m j=1 m j=1 X 1j (s, ω)db j (s, ω).. X nj (s, ω)db j (s, ω) Xij db j still remains a martingale, Because of lemma 1.7 we know that j moreover the single summands are one-dimensional Itô integrals. Continuation from L 2 [, T ] to H 2 [, T ] We want to enlarge the class of processes from L 2 [, T ] to a greater vector space H 2 [, T ]. This allows us to apply our instruments to a bigger class of stochastic processes than before. Notation H 2 [, T ] := H 2 ([, T ], Ω, F, {F t } t I, P ) := {(X t, F t )} t I, X t (ω) R {X t } t I is F t -adapted, Xt 2 dt < P -a.s. X H 2 [, T ] cannot be approximated by φ (n) S (elementary processes) as we did for processes in L 2 [, T ]. But they can be located by stopping times τ and with this sort of location we achieve that X := X τ L 2 [, T ]. The approximation for X follows analogous as for all processes in L 2 [, T ] Definition A F-measurable function τ : Ω [, T ] w.r.t. a filtration {F t } t I with A t := {ω Ω τ(ω) t} F t t I is called stopping time. Explanation With help of a stopping time it is possible to stop a stochastic process and hold it in its actual condition. { Xt (ω) for t τ X t τ (ω) := is a stopped process. X τ(ω) (ω) for t > τ If we define a new σ-algebra, an algebra of occurrences, till time τ as F τ := {A F A A t F t t I}, we will get the stopped filtration {F t τ } t I. Note that τ is F τ -measurable and F t τ F t. The restriction A t F t tells us that we are able to decide whether or not to stop the process. An example for a stopping time would be an investor who sells all his securities when they have reached a certain value. These stopping times enable us to locate stochastic processes as we see in the next definition. Let be I := [, ).

25 1.2 The Ito calculus Definition Let {(X t, F t )} t I be a stochastic process with X =. If {τ n } n N with ( ) τ n τ n+1 a sequence of stopping times with P lim τ n = = 1, such that n N { } X (n) t := X t τn, F t is a martingale n N, then we call X t a local t [, ) martingale and {τ n } n N is the localizing sequence. Now, we can localize { processes X H 2 [, T ]: } Define a stopping time τ n (ω) as τ n (ω) := T inf t T Xs 2 (ω)ds n and a stopped process X (n) t (ω) as X (n) t (ω) := X t (ω) χ {τn (ω) t}(t) It follows (without proof): X (n) t (ω) L 2 [, T ]. Then we denote the stochastic integral as I t [X] := I t [ X (n) ] for t τ n. Note that I t [X] is a local martingale by construction. Conclusion With this approximation from H 2 [, T ] over L 2 [, T ] by elementary processes we can solve the stochastic integral (1.1) for all processes in H 2 [, T ]. Thus, we presume that our processes belong to the much greater class of H 2 [, T ]. The Itô formula As we have seen solving stochastic integrals is not that easy. We need a significant tool which shows us a solution for some sorts of these integrals: the Itô formula. Before we reach at this theorem, we need some basic notations and definitions. Assumptions (Ω, F, P ) is a complete probability space, I = [, ) {F t } t I is a filtration fulfilling the usual conditions 1.3 {(B t, F t )} t I is a m-dimensional Brownian motion {K t } t I is a F t -adapted process with K t H 2 [, T ] K s < P -a.s. t I {L t } t I is a F t -adapted, m-dimensional process with L t (ω) := (L 1 (t, ω),..., L m (t, ω)) and j = 1,..., m L j (t, ω) H 2 [, T ] L 2 j (t, ω) < P -a.s. t I and

26 14 Stochastic analysis 1.21 Definition {(X t, F t )} t I is a Itô process assuming values in R, if a depiction for X t t I such as X t (ω) = X (ω) + = X (ω) + K s (ω)ds + K s (ω)ds + P -a.s. in which X (ω) is F -measurable. m j=1 L s (ω)db s (ω) L j (s, ω)db j (s, ω) Notation a) An n-dimensional Itô process X = ( X (1),..., X (n)) t has Itô processes X (i), i = 1,..., n, as components. b) For an Itô process X t we often use the differential representation: dx t = K t dt + L t db t. c) J := {X t (ω) X t (ω) is Itô process } 1.22 Definition Let X t, Y t J with X t = X + K s ds + L s db s and Y t = Y + N s ds + O s db s. X t, Y t := m j=1 is called quadratic covariation of X t and Y t. L j (s, ω)o j (s, ω)ds m X t := X t, X t := j=1 is the quadratic variation of X t. L 2 j(s, ω)ds Assume X t J, Y t : [, ) Ω R is F t -adapted. Then we define Y s dx s := Y s K s ds + Y s L s db s.

27 1.2 The Ito calculus Theorem (One-dimensional Itô formula) Let B t be a Brownian motion, X t J with X t = X + f : R R, f C 2 (i.e. f is continuous). For all t I we have K s ds + L s db s and f(x t ) = f(x ) + = f(x ) + f (X s )dx s f (X s )d X s ( f (X s )K s f (X s )L 2 s ) ds + P -a.s. f (X s )L s db s (1.5) Proof: see Korn [4], pp First, secure with help of a localization that all expectations are defined and the marginal processes can be transposed. Applying Taylor expansion, show that the occurring sum converge to integrals of the Itô formula. Remarks a) All integrals in (1.5) are defined. b) Once more: the differential representation for (1.5) : df(x t ) = f (X t )dx t f (X t )d X t Applications of theorem 1.23 a) X t = t X t = + 1ds + db s f(x t ) = f(t) with f C 2 f(t) = f() + f (s)ds b) X t = B t, f(x) = x 2, f() = f (x) = 2x, f (x) = 2 B t = + B 2 t = 2 K s ds + B s db s L s 1 dbs (remember that B = ) 2ds = 2 B s db s + t (cp. example 1.18) For a product of two Itô processes X t, Y t we can apply the Itô formula, too. The next theorem is an implication of the multidimensional Itô formula, see theorem A.8, appendix A.2.

28 16 Stochastic analysis 1.24 Theorem (Partial integration) Let X t, Y t J with dx t = K t dt + L t db t and dy t = N t dt + O t db t. X t Y t = X Y + = X Y + = X Y + X s N s ds + X s dy s + X s O s db s + Y s dx s + (X s N s ds + Y s K s + L s O s ) ds + d X, Y s Y s K s ds + Y s L s db s + (X s O s + Y s L s ) db s L s O s ds The Itô martingale representation theorem Notation A martingale M t assuming values in R w.r.t a Brownian filtration F t is called Brownian martingale. The next theorem will help us to construct the completeness of the market (chapter 2.3) and supports the deduction of portfolio optimization (chapter 3) Theorem (The martingale representation theorem) Let {(M t, F t )} t [,T ] be a Brownian martingale with E [ ] Mt 2 < t [, T ], B t a m-dimensional Brownian motion. [ ] T a F t -adapted process κ : [, T ] Ω R m with E κ(s, ω) 2 ds < with M t (ω) = M (ω) + κ(s, ω) t db(s, ω) Proof: see Korn [4], pp The proof is very technical and basically consists on the comparison of vector spaces and their equality. Remark a) κ is P λ unique b) For a local Brownian martingale theorem 1.25 is still valid (with a matching localization).

29 1.3 Solutions of stochastic differential equations Solutions of stochastic differential equations For modelling e.g. capital processes in chapter 2.2, we will need stochastic differential equations (= SDEs) and their solutions. The next theorem will help us to solve them Theorem Let {(B t, F t )} t I be a m-dimensional Brownian motion, σ j, j = 1,..., m, and b F t -adapted processes with i) σ j : [, T ] Ω R, with (t, ω) σ j (t, ω) b : [, T ] Ω R, with (t, ω) b(t, ω) ii) b(s) ds < t I σ j (s) 2 ds < t I and for j = 1,..., m Then the homogeneous SDE ( dp (t) = P (t) b(t)dt + P () = p m j=1 ) σ j (t)db j (t) (HSDE) has the unique solution P (t) = p exp (b(s) 1 2 m m σ j (s) 2 )ds + j=1 j=1 σ j (s)db j (s) Proof i) P () = p exp() = p ii) Let be m = 1 (otherwise we apply the multi-dimensional Itô formula, see theorem A.8, Appendix A.2). Z t := + f(z) := p exp(z) ( b(s) 1 2 σ(s)2) ds + σ(s)db s

30 18 Stochastic analysis We apply the Itô formula 1.23: f(z t ) = p + + [ f (Z s ) (b(s) 12 ) σ(s) ] f (Z s )σ(s) 2 ds f (Z s )σ(s)db s [f(z s ) = f (Z s ) = f (Z s ) = p exp(z s ) = P (s)] P (s) = p + P (s)b(s)ds + P (s)σ(s)db s dp (s) = P (s) (b(s)ds + σ(s)db s ) P (t) = f(z t ) = p exp (b(s) 1 t 2 σ(s))ds + σ(s)db s For the P λ-uniqueness of the solution we define Z t := P (t) 1 and assume that P (t) is another solution for (HSDE). Z t = P (t) 1 = p exp ( ) 1 2 σ(s)2 b(s) ds + dz t = Z t (( 1 2 σ(t)2 b(t) σ(t)2 ) ds σ(t)db t ) = Z t (( σ(t) 2 b(t) ) ds σ(t)db t ) σ(s)db s We apply the partial integration, theorem 1.24, and get P (t)z t = 1 + P (s)z s (( σ(s) 2 b(s) + b(s) σ(s) 2) ds + (σ(s) σ(s)) db s ) P (t)z t = 1 P (t) = 1 Z t = P (t) With p = P () = P () the solution is unique. But more complicated SDE will occur. For the general case of theorem 1.26 we have the following observation.

31 1.3 Solutions of stochastic differential equations Theorem (Variation of constants) Assumptions: σ j, b and {(B t, F t )} t I as in theorem 1.26 x R β, S j F t -adapted, real valued processes satisfying β, S j C, ( β(s) )ds < and Then the SDE dx(t) = (b(t)x(t) + β(t)) dt + X() = x (S j (s)) 2 ds < P -a.s. t I and j = 1,..., m m (σ j (t)x(t) + S j (t)) db j (t) (SDE) j=1 has the unique solution w.r.t. λ P {(X t, F t )} t I with: X(t) = R(t) x + β(s) m R(s) m σ j (s)s j (s) ds + j=1 j=1 S j (s) R(s) db j(s) in which R(t) = exp is the unique solution of the HSDE (b(s) 1 2 σ(s) 2 )ds + dr(t) = R(t) ( b(t)dt + σ(s) t db t ) R() = σ(s) t db s For proof and annotations compare Korn [4], pp Analogous to theorem 1.26; apply the partial integration for X t. Now, we are ready to model stock prices and other significant properties of the market. The presented (especially stochastic) tools form the basis for a mathematical depiction of the financial world.

32

33 21 2 A model of the financial market We want to describe an economic market in a mathematical way. Therefore, we need to make assumptions and we have to admit restrictions in our model to be able to apply the presented instruments and tools. During the depiction we will collect all these assumptions and restrictions and summarize them at the end of the chapter into an overview. As we mentioned in the introduction, our portfolio only contains stocks and bonds. 2.1 Modelling prices To get to know the approach of modelling prices we begin with a risk-free asset: the bond. First, what is the meaning of risk-free? Risk-free tells us that the security does not depend on chance or random which means that the future price of a bond is predictable for the investor. By analyzing former performances and future expectations he gets an expected yield or an expected price for the asset. The bond price Let P (t) be the price of the bond at time t, t [, T ], T <. We buy at t = and pay P () := p. Suppose after one period (for example one year, t = 1) the first expected, constant interest rate r (r R + ) is payed. Our capital rises to P (1) = p + r p = (1 + r) p. Assume now that this interest rate is credited n-times a period, then we have (binomial coefficient): P (1) = ( p + r ) ( n p + p + r n p ( )( n r = p (1 + 1 n ( = p 1 + r ) n n ) )( r n ( n 1 ) ( p + r n p )( r n ) n 1 + ( ) n ) r n )( ) n 1 r n For n follows: P (1) = p exp(r 1) or rather P (t) = p exp(rt) for t I (2.1) But what if the interest rate r is not constant? We know that some yields of bonds are floating in dependence of time, but they are no subject of chance. Hence, we write (2.1) for a continuous-time (i.e. non-constant, time dependent and continuous, means r C ) rate of interest: P (t) = p exp r(s)ds for t I (2.2)

34 22 A model of the financial market We note that (2.2) is the unique solution for the differential equation dp (t) = P (t)r(t)dt with P () = p, t I (DE) P (t) = p + P (s)r(s)ds With (2.1) or rather (2.2) we have found a mathematical depiction of the bond price. Stock prices What are the similarities and differences of a stock price compared to a bond price? The stock price is driven by chance; the investor takes over a certain risk when he buys stocks. This risk has to be compensated with a higher interest rate. This rate can be deducted from the former performances and data of the stock and from the expectations for the asset s future. Thus, this part of the stock price can be determined and is consequently a sort of expected price. Concluding, a stock price is similar to the bond one, but cannot be identified in the same manner applied before. Under (2.1) we chose the log-linear-approach. Assume, n stocks are given. The i-th stock price at time t is P i (t), b i the expected rate of return and P i () = p i the actual price at time t =. For a bond we have (compare (2.1)): ln(p (t)) = ln(p ) + r t Hence, we derive the following price for stock i: ln(p i (t)) = price at t = {}}{ ln(p i ) + return rate {}}{ b i t } {{ } expected price + chance Let us describe chance (often called white-noise or white-noise-effect ). What demands do we have on the random influencing stock prices? i) It has to be without tendency, mathematically this means E[ chance ] =. ii) It has to be without memory, i.e. chance at time t is independent of chance at time s, s < t, s, t I. iii) It depends on time t. iv) It describes the sum of the deviation of the real price P i (t) from the calculated or rather expected price ln(p i ) + b i t on [, T ]. If these deviations are independent, we can assume that chance is N(, σ 2 t), σ R +, distributed which can be concluded by the central marginal theorem (see Bauer [6] or Michel [1]).

35 2.1 Modelling prices 23 Then, a definition of chance is suggested: Y (t) := ln(p i (t)) ln(p i ) b i t Y (t) fulfils i) and iii). Moreover it shall be that Y (t) Y (s) depends on (t s) and is independent of Y (u) for u s. This fulfils i),ii) and iv). In other words Y (t) Y (s) N(, σ 2 (t s)). All these properties of chance are held by the Brownian motion {(B t, F t )} t I with filtration {F t } t I. But what about the Markowitz volatility? The solution for this is obvious. As we deducted the expected interest rate b i, it is possible to get the volatility of a stock i from its former data, variation and future expectation. It is suggested that this can be described by the variance σ i of the underlying. But we have to pay attention: stocks are correlated among themselves. The volatility σ ij expresses the dependence of asset i from asset j, σ ii is then the calculated variation. All this leads to the following equation: ln(p i (t, ω)) = }{{} ln(p i ) }{{} + bi t }{{} + σ ii B t (ω) }{{} actual price price at t= expected yield volatility connected with chance ( ) P i (t, ω) = p i exp bi t + σ ii B t (ω) (t, ω) I Ω (2.3) Because (2.3) is only for one stock, we have to modify for the general case to m P i (t, ω) = p i exp bi t + σ ij B j (t, ω) for i = 1,..., n (2.4) j=1 in which B(t, ω) = (B 1 (t, ω),..., B m (t, ω)) t is a m-dimensional Brownian motion. Remarks i) In the case n = m σ ij represent the coherence between stock i and j. ( ) ii) We note that m σ ij B j (t, ω) is N, m σij 2 t distributed. j=1 j=1 ( ) ln(p i (t, ω)) N ln(p i ) + b i t, m σij 2 t iii) Abbreviation: P i (t, ω) := P i (t) We define b i := b m i σ ij (s) 2, i = 1,..., n, and apply theorem 1.26 to (2.4). j=1 j=1

36 24 A model of the financial market We see that m P i (t, ω) = p i exp bi t + σ ij B j (t, ω) j=1 for i = 1,..., n is the unique solution for the HSDE m dp i (t) = P i (t) b i dt + σ ij db j (t) j=1 P i () = p i for i = 1,..., n (2.5) With (2.5) we have achieved the equation of stock prices, depicted as Itô processes. 2.1 Lemma (Properties of stock prices) Let b i := b i m σij 2 j=1. With (2.4) we have i) E[P i (t)] = p i exp(b i t) ( ( ) ) ii) V ar(p i (t)) = p 2 i exp(2b m i t) exp σij 2 t 1 j=1 ( m ( iii) Y t := a exp cj B j (t) 1 2 σ2 ij t)), j=1 a, c j R, j = 1,..., m is a martingale with E [Y t ] = 1. Proof: see appendix B. W.l.o.g. set m = 1, use the definition of P i (t) and the properties of B t (i.e. B t N(, t) and B t B s independent of F s ). Compute the martingale condition E[Y t F s ] = Y s. Using the definition of b i we confine that m P i (t, ω) = p i exp(b i t) exp σ ij B j (t, ω) 1 2 σ2 ijt P i (, ω) = p i i = 1,..., n j=1 } {{ } ϱ (2.6) The figure shows four possible price developments of a single stock received by four different paths of the Brownian motion.

37 2.1 Modelling prices 25 stock price P t with P = P (ω ) t 1 P t (ω 2 ) P t (ω 3 ) P (ω ) t 4 E[P t ] 5 t= t=t/2 t=t t Figure 2.1: Possible stock prices and the stock price expectation Notation i) p i exp(b i t) is the expected stock price, b = (b 1,..., b n ) t is the vector of expected (anticipated) returns. ii) ϱ models the deviation from the expected price ( chance ), σ σ 1m σ := is the matrix of volatility σ n1... σ nm iii) P i (t, ω) : [, T ] Ω R is called geometric Brownian motion with drift b i and diffusion or volatility σ i := (σ i1,..., σ im ) However, we see that this model is not continuous-time: returns and volatilities do not depend on time which is not realistic in an economic market. If we assume that the expected returns b i (t) and volatilities σ ij (t) are time-dependent, integrable and continuous (i.e. b, σ C ), then we get for the bond and stock prices with b i (s, ω) = b i (s, ω) P (t, ω) = p exp r(s, ω)ds m j=1 σij 2 (s, ω): P (, ω) = p P i (t, ω) = p i exp P i (, ω) = p i b i (s, ω)ds + m σ ij (s, ω)db j (s, ω) j=1 i = 1,..., n and t I (2.7)

38 26 A model of the financial market Demands on our price model r(t) = r(t, ω), b(t) = b(t, ω) = b 1 (t, ω).. b n (t, ω), σ 11 (t, ω)... σ 1m (t, ω) σ(t) = σ(t, ω) = σ n1 (t, ω)... σ nm (t, ω) are F t -adapted and regular bounded processes, r, b and σ C. σ(t)σ(t) t : K > with x t σ(t)σ(t) t x Kx t x which means that σ is uniformly positive definite. x R and t I With this we apply theorem 1.26, chapter 1.3, and write (2.7) as Itô processes. dp (t) = P (t)r(t)dt P () = p m dp i (t) = P i (t) b i (t)dt + σ ij (t)db j (t) P i () = p i i = 1,..., n j=1 (2.8) Remarks The bond is not risk-free anymore, but its dependence on chance is severely restricted because r(t, ω) is regular bounded. In practice this comes closer to reality as the suggestion that the bond is a complete risk-free asset. (t, ω) [, T ] Ω describes the market situation. 2.2 Portfolio processes What kind of possibilities does an investor have to act or to react on the market? First, he has a certain amount of money which he wants to invest (starting capital). Then, after his first investment, he can redistribute his capital. This means to sell assets and to invest in other securities. On the other hand, he can consume parts of his property by selling without reinvestment. A negative consumption would be, if the investor added money to his existing portfolio. We will not regard this case. Besides the actions of the investor, we have to formulate master conditions for our market. Some are obvious: The investor may not have insider information. The investor is not able to influence the price trend of assets (maxim of the small investor )

39 2.2 Portfolio processes 27 At the beginning (t = ) the investor has a certain starting capital (x >, initial value). This starting capital has to be invested (normally completely) in n stocks and one bond. And some are necessary for our theory: Portfolios are self-financing, i.e. every change concerning the investor s capital results from consumption or redistribution. Assets are divisible by any number. There are no costs of transaction. Short sellings and credits are allowed: A negative share of a bond symbolizes a credit. The interest rate r is for an investment in a bond as well as for a credit (compare the next point). A negative share of a stock means the investor owes to another investor these stocks (short sellings). The market is complete, every investor gets the same conditions. The information about securities are obtainable for everyone. Interest rate r for credits and bonds is equal for every participant (see previous aspect). This implies that r(t, ω) only depends on the market situation (t, ω) [, T ] Ω and not on the credit standing status of the investor. Market imperfections are excepted. An introduction to the expression self-financing Let us develop our theory by regarding an example. Let x R + be the investor s starting capital to be invested for two periods of time, i.e. t {, 1, 2}. The capital process X(t) describes the investor s wealth at time t. His capital rises in the case of win and it descends in the case of loss or consumption which is understood as the consumption process C(t) with C() =. Suppose we only trade one bond and one stock. Their prices are defined as P (t) and P 1 (t), see chapter 2.1. At time t = the investor determines the shares of the starting capital x to be put into the bond (:= ϑ ()) and into the stock (:= ϑ 1 ()). This leads to the following approach: X() = ϑ () P () + ϑ 1 () P 1 () = x (1) After the first period (t = 1), if the investor consumes parts of his wealth, will say C(1) >, we have X(1) = ϑ () P (1) + ϑ 1 () P 1 (1) C(1) (2) In addition to that he can redistribute his portfolio which means: X(1) = ϑ (1) P (1) + ϑ 1 (1) P 1 (1) (3) We regard this in an abstract way. What is X(1) from a general point of view? X(1) = starting capital + win/loss bond + win/loss stock - consumption

40 28 A model of the financial market Nevertheless, it is also, see equation (3): X(1) = share bond t=1 bondprice t=1 + share stock t=1 stockprice t=1 From equations (1) - (3) we obtain: X(1) = x + ϑ ()(P (1) P ()) + ϑ 1 ()(P 1 (1) P 1 ()) C(1) (4) X(1) = ϑ (1) P (1) + ϑ 1 (1) P 1 (1) (5) Analogous we get for t = 2: X(2) = x ϑ (i 1)(P (i) P (i 1)) i=1 2 ϑ 1 (i 1)(P 1 (i) P 1 (i 1)) i=1 2 C(i) (6) X(2) = ϑ (2) P (2) + ϑ 1 (2) P 1 (2) (7) The fact that equations (6) and (7) or rather (4) and (5) are equal is called self-financing. This makes sense for a discrete-time model, but we need a continuous-time one which requires that the investor may trade or consume at every time t I. However, this means that the occurring sums in (6) or (4) merge into integrals (the differences are getting infinitesimal small). The result is X(t) = x + ϑ (s)dp (s) + i=1 ϑ 1 (s)dp 1 (s) C(s)ds (2.9) We recall that P i, i = 1, 2, are Itô processes and we have to make restrictions such that the integrals in (2.9) exist. With this in mind we specify as follows: 2.2 Definition a) A trading strategy is a F t -adapted process ϑ : [, T ] Ω R n+1, ϑ(t) := ϑ(t, ω) := (ϑ (t, ω),..., ϑ n (t, ω)) t such that T i) ϑ (s) ds < P -a.s. and ii) T (ϑ i (s)p i (s)) 2 ds < P -a.s. for i = 1,..., n. b) x := n ϑ i () p i is called initial value of ϑ. i= c) Let C : [, T ] Ω R +, C(t) := C(t, ω) be a F t-adapted process T with C(s)ds < P -a.s.. Then C(t) is a consumption process.

41 2.2 Portfolio processes 29 Annotations That a trading strategy is F t -adapted is the conversion of the demand that the investor does not have any insider information. His decision to sell or to buy only depends on the market situation (t, ω) [, T ] Ω. The initial value x is (normally) completely invested in the portfolio. ϑ R n+1 is held, because in the cases of credits and short sellings its value can be negative. We remark that with C(t) t I no added money is admitted. 2.3 Definition Let x R + be an initial value and ϑ a trading strategy. a) Then X : [, T ] Ω R, X t := X(t) := X(t, ω) := n i= is the wealth or capital process w.r.t ϑ. ϑ i (t, ω) P i (t, ω) := n ϑ i (t) P i (t) b) For X t > P -a.s. π : [, T ] Ω R n, π(t) := π(t, ω) := (π 1 (t, ω),..., π n (t, ω)) t with i= π i (t, ω) := ϑ i(t, ω)p i (t, ω) X(t, ω) is called portfolio process w.r.t. ϑ. i = 1,..., n The capital process X t allows to determine directly the actual value of the investor s portfolio. As we introduced the term self-financing, we came to the conclusion that actual wealth = initial value + win/loss assets - consumption must be valid. We formalize this in the following definition. 2.4 Definition Let ϑ be a trading strategy, C a consumption process and X the accompanying capital process. a) Then (ϑ, C) is self-financing, if the following equation is held P -a.s. t I: X(t) = x + n i= We remember (2.8) and realize that ϑ i (s)dp i (s) C(s)ds (2.1) X(t) = x + + n ϑ (s)p (s)r(s)ds + i=1 n i=1 m ϑ i (s)p i (s)σ ij (s)db j (s) j=1 ϑ i (s)p i (s)b i (s)ds C(s)ds (2.11)

42 3 A model of the financial market b) If (ϑ, C) is self-financing and X(t) P -a.s. t I, a portfolio process π is called self-financing portfolio process (π, C) w.r.t. (ϑ, C). Remarks X(t) and all integrals in (2.1) and (2.11) are defined and exist, because of the assumptions for C, ϑ i, P i, i =,..., n, in definition 2.2 and for r, b, and σ in chapter 2.1. If X(t) and P i (t), i = 1,..., n, are known, we have (ϑ, C) (π, C). Furthermore it is X n ϑ i P i (1 π t e n i=1 ) = X = ϑ P X To get X(t) in form of a SDE, we compute (compare (2.1) and (2.11)): Notation The SDE dx t = i ϑ i dp i Cdt = ϑ P rdt + i = (1 π t e n )X t rdt + i ϑ i P i b i dt + i ϑ i P i σ ij db j Cdt j π i X t b i dt + i π i X t σ ij db j Cdt = (1 π t e n )X t rdt + X t π t bdt + X t π i σ i db Cdt i }{{} =X t π t σdb = X t π t [(b e n r)dt + σdb] + (X t r C)dt j dx t = (X t r(t) C(t))dt + X t π(t) t [(b(t) r(t)e n )dt + σ(t)db t ] X = x (CE) is called capital equation (CE). We see that r(t), b(t), σ(t) and C(t) fulfil the assumptions of theorem 1.27 (variation of constants). To get an unique solution for (CE), it is required that π i (t) 2 dt < P -a.s., i = 1,..., n (A)

43 2.2 Portfolio processes 31 Notation (compare with definition 2.4 b) If (CE) has an unique solution X t := X π,c (t, ω) := X π,c (t) with (X t π i (t)) 2 dt < P -a.s., i = 1,..., n (B) π(t) is called self-financing portfolio process and we denote X t as accompanying or corresponding capital process. Let us examine the restrictions (A) and (B) and ii) in definition 2.2. First, we remark that T T (X t π i (t)) 2 dt < (ϑ i (t)p i (t)) 2 dt < Then, we know that X t is continuous. This implies that with also T (X t π i (t)) 2 < is valid (in other words: (A) (B)). T π i (t) 2 dt < If we assume (A), our (CE) is solved by a capital process X t with X t >. This results from the explicit solution for (CE) (compare theorem 1.27). Whereas under restriction (B) for (CE) the solution X t is in R, i.e. the investor could go bankrupt (X t = ) or get into a debt position (X t < ). 2.5 Definition Let (ϑ, C) / (π, C) be a self-financing trading strategy ϑ / portfolio process π with a consumption process C and x > as initial value. If the capital process X t holds X t P -a.s. t I, we call X t admissible with initial value x. Further, we define D ϑ (x) := {(ϑ, C) (ϑ, C) is self-financing with admissible X t, X() = x} D π (x) := {(π, C) (π, C) is self-financing with admissible X t, X() = x} 2.6 Example Let x > be the initial value. Suppose the investor does not want to trade, which means that the shares of securities stay constant (i.e. ϑ(t) ϑ). Moreover, he does not consume any of his capital during the investment, so it is C(t). Then, we have with π(t) = ϑ P (t) X(t) and X() = x: dx(t) = X(t)r(t) + X(t)π(t) t [(b(t) r(t)e n ) dt + σ(t)db t ] dx(t) = X(t) [ r(t) + π(t) t b(t) π(t) t r(t)e n] dt + X(t) π(t) t σ(t) db t }{{}}{{} b(t) σ(t)

44 32 A model of the financial market With theorem 1.26 we get as solution: X(t) =x P (t) P (t) = exp exp r(s) + π(s) t (b(s) r(s)e n ) 1 2 π(s)t σ(s) 2 ds π(s) t σ(s)db s On implication is surely that X t t I (ϑ(t), C(t)) = (ϑ, ) D ϑ or (π(t), C(t)) = (π(t), ) D π. 2.3 The complete market In the previous section we introduced the term complete market in an economical sense. In this part we deal with the completeness of the market in a mathematical way. Normally, the investor wants to maximize his yields or he wants to live according to a consumption process determined by him in advance. He may also aim for certain predestined terminal value in the end of the investment. Thus, we have to find a discount factor, especially for the last two aspects. It must allow to calculate back from the terminal aims or occurrences to any time t [, T ] of the beginning investment. As the market is complete in an economical way, we conclude that one part of the discount factor must be the interest rate r of the bond or credits. Hence, we have δ 1 (t) := δ 1 (t, ω) := exp Further, we define r(s, ω)ds = exp ξ(t) := ξ(t, ω) := σ(t, ω) 1 (b(t, ω) r(t, ω)e n ) δ 2 (t) := δ 2 (t, ω) := exp = σ(t) 1 (b(t) r(t)e n ) = exp δ(t) := δ 1 (t) δ 2 (t) ξ(s, ω) t db s (ω) 1 2 ξ(s) t db s 1 2 ξ(s) 2 ds r(s)ds ξ(s, ω) 2 ds The next figure shows four possible paths of δ(t, ω) with constant market coefficients σ, b and r R.

45 2.3 The complete market δ(t,ω 1 ) δ(t,ω 2 ) δ(t,ω 3 ) δ(t,ω ) 4 δ(t,ω) t Figure 2.2: The discount factor δ(t, ω) Properties of the discount factor δ First, we remark that δ(t) > t I. As r, b and σ are continuous, regularly bounded and F t -adapted, δ(t) is continuous and F t -adapted as well. Moreover, we see that δ(t) has the same structure as the unique solution P (t) in theorem Hence, we can describe δ(t) as a SDE or as Itô process: dδ(t) = δ(t) ( r(t)dt + ξ(t) t db t ) Explanation δ() = 1 (2.12) ξ(t) is a relative market risk premium. For the better understanding we regard the case n = 1 and constant processes r, b and σ. This leads to ξ b r σ. We recognize this expression: it is the Sharpe ratio used for the capital market line (CML, deduced by Tobin from Markowitz theory). Obviously, ξ indicates the premium for the risk to invest into stocks instead of investing into the bond. Compared to the modelling of stock prices it symbolizes chance for the discount factor. For δ(t) or rather δ 2 (t) we remember the deduction of the stock prices. In equation (2.6) we modelled stock prices and annotated that the expression ϱ stands for the deviation of the expected price. As δ 1 (t) represents a sort of expected discount factor, we add a discount factor depending on chance (= δ 2 (t)). The solution of 2.12 explains the special form of δ 2 with the additional term 1 2 ξ(s) 2 ds. Thus, δ(t) can be regarded as a discount factor depending on the market situation (t, ω) [, T ] Ω. δ(s, ω) describes the actual value of money means payed at time s in the market condition ω.

46 34 A model of the financial market Now, with help of δ(t), we are able to present an important theorem which shows an astonishing feature of the just developed market. 2.7 Theorem (Completeness of the market) Let be π a portfolio process C a consumption process x R + the initial value (π, C) D π (x) and δ(t) the above-mentioned discount factor i) For the accompanying capital process X t we have: E δ(t)x t + δ(s)c(s)ds x t I ii) Let Q be a F T -measurable [ random variable, Q], and let the initial T value X hold: x := E δ(t )Q + δ(s)c(s)ds a portfolio process π(t), t I, with π D π (x) such that the corresponding capital process satisfies X T = Q P -a.s. Proof: see appendix B. Show that δ(t), [ X t J, apply theorem 1.24 to δ(t) X t. Define T X t := δ(t) 1 E δ(s)c(s)ds + δ(t )Q F t ], t [ ] T M t := δ(t) 1 E δ(s)c(s)ds + δ(t )Q F t and show that δ(t)x t + δ(s)c(s)ds = M t. Furthermore, check that M t is a martingale, apply theorem 1.25, determine κ and show that (π, C) D π (x). To be mentioned: π is the unique portfolio process except from P λ - equivalence.

47 2.3 The complete market 35 Interpretation of theorem 2.7 First, we separate [ ] E δ(t)x t + δ(s)c(s)ds into ( ) E [ [δ(t)x t ] and t ] ( ) E δ(s)c(s)ds. If we interpret δ(t) as a suitable discount factor, we see that ( ) indicates the expected, discounted capital at time t, whereas ( ) stands for the expected, discounted and average consumption. For t = T, X T symbolizes the terminal capital. Thus, E [δ(t )X T ] }{{} discounted, terminal capital + E δ(s)c(s)ds }{{} discounted consumption manifests the needed initial value to achieve desired aims of the investor, e.g. a given terminal capital or living according to certain consumption. Therefore, part i) restricts the investor s wishful thinking concerning consumption and terminal wealth. If we regard goals permitted by part i), part ii) tells us that they can be realized with help of these initial endowments. In other words, if we predetermine a consumption process and a terminal wealth satisfying the assumptions [ E ] δ(s)c(s)ds δ(t)x t + x and [ ] T E δ(t )Q + δ(s)c(s)ds = x we can find a portfolio process with (π, C) D π (x). Hence, there is one remarkable feature of this theorem: We can achieve every desired terminal capital with help of a suitable, self-financing portfolio process, if we just have enough starting capital. A market with this property is called complete market. 2.8 Example X t := δ(t) 1 corresponds to the strategy (π(t), C(t)) := ((σ(t) 1 ) t ξ(t), ) with X = 1.

48 36 A model of the financial market Proof 1. X = δ() 1 = 1 ( 2. δ(t) 1 = exp r(s) ξ(s) 2 ds + ξ(s) t db s ) 3. We remember example 2.6 (C(t) ) and use X t = δ(t) 1 or rather x = 1 and π(t) = (σ(t) 1 ) t ξ(t): X t = exp exp exp = exp ( ) ( r(s) + ((σ(s) 1 ) t ξ(s)) t (b(s) r(s)e n ) ) ds exp exp = exp We see ( ) = ( ). 1 ( (σ(s) 1 ) t ξ(s) ) t 2 σ(s) ds 2 ( (σ(s) 1 ) t ξ(s) ) t σ(s)dbs =ξ(s) {}}{ r(s) + ξ(s) t σ(s) 1 (b(s) r(s)e n ) }{{} = ξ(s) 2 1 ξ(s) t σ(s) 1 σ(s) 2 ds 2 ξ(s)σ(s) 1 σ(s)db s (r(s) ξ(s) 2 )ds + ξ(s) t db s ( )

49 2.4 Assumptions for the complete market Assumptions for the complete market A small overview concerning the assumptions required for our financial market model. Economic premises n stocks and one bond (no other securities) maxim of the small investor no insider knowledge short sellings and credits possible no transaction costs no market imperfections, a complete market in the economical way Mathematical requirements self-financing portfolio processes π admissible capital processes X t (X t P -a.s. t I, X = x > ) a complete market in the mathematical way assets divisible by any number restrictions for r, b, σ and the consumption process C: r(t, ω) F t -adapted, R-valued, regularly bounded process, r C b(t, ω) F t -adapted, R n -valued, regularly bounded process, b C σ(t, ω) F t -adapted, R n m -valued, regularly bounded process, σ C, uniformly positive definite C(t, ω) F t -adapted, R n -valued process with C(t)dt < I = [, T ], T <, finite time horizon B(t, ω) is a m-dimensional Brownian motion, F t the Brownian filtration (t, ω) [, T ] Ω describes the market situation at time t (Ω, F, P ) is a complete probability space n = m, the dimension of the Brownian motion corresponds to the number of stocks

50

51 39 3 Portfolio optimization In the previous chapter we presented a model of an economical and mathematical complete market. It has a fixed time horizon with I = [, T ] and is continuous in time (i.e. trading and consumption is allowed at any time t I). Moreover, it comprises n + 1 securities, one bond and n stocks. Remember the price description (2.7) driven by a n-dimensional (n = m) Brownian motion and influenced by the rates of return r(t) for the bond (possibly randomly), b i (t), i = 1,..., n, for the stocks (randomly fluctuating) and by the volatility coefficients σ ij (t), i, j = 1,..., n with its assumptions (see chapter 2.4) and the resulting settings and properties. The complete market model will allow us to develop a method to optimize our portfolio or respectively portfolio processes: the martingale approach. The martingale approach was introduced in the 198ies in different versions by Cox and Huang, Pliska, Karatzas et al, compare p.e. Karatzas, Lehoczky and Shreve [11]. The other main approach to solve the later presented unconstrained portfolio problem (P ) was developed by Merton in 1969: the stochastic control approach. As Merton was one of the first introducing and solving the problem, (P ) is sometimes called Merton s problem. The basis for the stochastic control approach is the stochastic control theory. Here, a solution for (P ) is computed by solving a Hamilton-Jacobi-Bellmann (HJB) equation in two steps: First, an optimal portfolio and consumption processes are searched in dependence of the unknown optimal expected utility. Putting this solution into the (HJB) equation, the result is a non-linear partial differential equation. It can be solved under special conditions while it is hard to get a solution in the general case. In addition to that the resulting numerical problem often has no explicit solution. For more information refer to Merton [12, 13]. However, we direct our focus to the martingale method. 3.1 The continuous-time portfolio problem To get to our portfolio problem we take a closer look at the investor s problem: He is endowed with an initial value x. Now, he must choose strategies for his investment and his consumption in dependence of the stocks price trends. This means he has to determine what share of which stock at which time he holds to get a terminal value as high as possible and how much capital he consumes at which time. In other words he wants to maximize the utility of terminal wealth and consumption. Thus, we need to find a measure for utility - the utility function.

52 4 Portfolio optimization Let us consider the problem from the economic point of view: First, it is obvious that with additional wealth or consumption the investor has more utility. Mathematically, this implies that the utility function must be strictly increasing. Further, it would be logical (in an economic sense) that the additional profit decreases the higher the unit of utility (i.e. consumption/wealth x) is. Viceversa, the loss of profit increases the lower the unit of utility is. In economic terms: the marginal profit in x = is infinite ( a little bit is better than nothing ), whereas it equals zero in x = (saturation effect). Hence, we denote as follows: 3.1 Definition i) Let u : (, ) R be a strictly concave function, u C 1, such that (a) u () := lim x u (x) = (b) u ( ) := lim x u (x) = Then, u is a utility function. ii) A function U : [, T ] (, ) R, U C,1 is called utility function in the sense of i), if U satisfies (a) U t ( ) is utility function for all t I and (b) U (x) C for fixed x R + Remarks For a detailed description of properties of utility functions read appendix A.3. With i) (a) and (b) the requirements for the marginal profit are satisfied ( a little bit is better than nothing and the saturation effect). 3.2 Example Examples for utility functions are u(x) = ln(x) u(x) = x U t (x) = exp( αt) u(x), α >, for a utility function u in the sense of definition 3.1 i). U t (x) = exp( t) ln(x)

53 3.1 The continuous-time portfolio problem 41 Figure 3.1: Utility function U t (x) = exp(.5t) x We pointed out that the investor would like to maximize the utility of the terminal value and consumption, in formula this means: U1 s (C(s))ds + U 2 (X T ) max! }{{} }{{} utility of terminal value utility of consumption in which U 1, U 2 are utility functions in sense of definition 3.1. However, we have to consider that we deal with random variables, that is why we use [ ] T E U1 s (C(s))ds + U 2 (X T ) max! We know that X t has a consumption process C(t) and a portfolio process π(t). And as we are in the complete market settings, it yields (π, C) := (π(t), C(t)) D π (x) with x as initial value. Then, we get as provisional result ( P ) max Ψ(x, π, C) = E s.t. (π, C) D π (x) [ T U s 1 (C(s))ds + U 2 ( X x,π,c (T ) )] Normally, we would restrict the problem ( P ) s.t. the occurring expectation is finite. But: the investor s aim is to gain as much (utility) as he can. Thus, a strategy with infinite utility would be the most desirable one. We would exclude an optimal strategy. Consequently, we define: Dπ(x) := (π, C) D π(x) E (U1 s (C(s))) ds + ( ( U 2 X x,π,c (T ) )) <

54 42 Portfolio optimization We see a negative, infinite utility is excluded whereas a positive, infinite utility is allowed for U 1, U 2 we have D π(x) = D π (x), otherwise D π(x) D π (x) With this we obtain the continuous-time problem for the initial value x R + : [ T ( max Ψ(x, π, C) = E U1 (P ) s (C(s))ds + U 2 X x,π,c (T ) )] s.t. (π, C) Dπ(x) 3.3 Definition (Portfolio problem) The optimization problem (P ) is called (unconstrained) portfolio problem or Merton s problem. Over the years one has defined many sorts of portfolio problems. Markowitz began with the mean-variance-approach, others admit transaction costs or allow an infinite time horizon, for example. In this thesis the depiction and solution of (P ) are severely depending on the complete market settings. 3.2 The martingale approach The main idea of the martingale approach is a decomposition of the dynamic portfolio problem (P ) into a static optimization problem (S) and a representation problem (R). In the static problem (S) we determine an optimal consumption and terminal wealth each geared to the needs or wishes of the investor. Then, in the representation problem (R), we compute a portfolio process π and a consumption process C which coincide with the solution of (S). Significant for this approach is the fact that we already know the optimal solution for (P ) (step 1: solution of (S)), before we have found its representation in form of processes (step 2: solution of (R)). Advantages and disadvantages A negative aspect is surely that the martingale approach is strongly restricted to the completeness of the market, mathematically and economically (see chapter 2.4). Some of these assumptions are not really realistic (e.g. market imperfections), but the complete market settings form the basis for the martingale approach. On the other hand this leads to several advantages: + non-constant, time- and random-dependent market coefficients + the preceding point implies a continuous-time market model, in particular this allows a continuously (negative) investment and consumption at any time t I.

55 3.2 The martingale approach 43 + general utility functions: the investor can freely choose the utility functions w.r.t. its definition 3.1. The choice of U 1 and U 2 reflects the investor s investment behaviour: he is able to control which part (terminal wealth or consumption) he emphasizes or neglects. + If the solution of (P ) has a non-closed form, the Monte-Carlo-simulation as numerical computation often delivers an explicit solution. We will not discuss this case. (+) For some problems it might be helpful to have an infinite time horizon I = [, ], but in most cases a finite time period I = [, T ], T <, is more suitable - a decision of the investor or the reader. A decomposition based on the complete market We decompose the problem (P ) max Ψ(x, π, C) = E (P ) s.t. (π, C) Dπ(x) [ T into the static problem (S) max (S) Ψ(Q, C) = E s.t. (Q, C) S in which C s := C(s) and { S := U s 1 (C s )ds + U 2 ( X x,π,c (T ) )] [ T U s 1 (C s )dt + U 2 (Q) (Q, C) Q, C, Q F T -measurable, C F t -adapted, E E δ(s)c s ds + δ(t )Q x ( ), (U1 s (C s )) ds + ( ( U 2 X x,π,c (T ) )) } < ( ) and the representation problem (R) { Find (π (R), C ) Dπ(x) with X x,π,c (T ) = Q s.t. Q solves (S) ]

56 44 Portfolio optimization Explanation of the decomposition (P ) (S) and (R) First, we see that C is a consumption process as (π, C) S. Moreover, if (π, C) D π(x), theorem 2.7 (Completeness of the market) guarantees that the corresponding capital process X t := X x,π,c (t) satisfies E δ(s)c s ds + δ(t)x t x for t I (3.1) especially, X T holds E δ(s)c s ds + δ(t )X T x (3.2) Thus, Q := X T and C s = C(s) yield ( ). In addition to that (π, C) D π(x) implies that X T, C and ( ) are valid. Now, theorem 2.7 ii) tells us that there exists a pair (π, C ) D π (x) with X x,π,c (T ) = Q P -a.s (3.3) As Q fulfills ( ) and is F T -measurable, we even have (π, C ) D π(x). This implies that D π(x) = S under the assumption of the completeness of the market model. Hence, (P ) and (S) have the same optimizer and this optimizer generates a solution for the representation problem (R). Please note that the completeness of the market model is evident for this decomposition and consequently for the martingale approach (see (3.1) to (3.3)). Later, we will see that the optimizer (Q, C ) of (S) yields E δ(s)cs ds + δ(t )Q = x Furthermore, a solution Q can always be found with e.g.: Q := x E[δ(T )] and C s s I P -a.s A heuristic optimization The Lagrangian multiplier method (see appendix A.4) will help us to solve the problem (P ) respectively problems (S) and (R) in a heuristic way. It is heuristic because we imitate this method and besides we treat random variables as (fixed) variables. In the next section 3.3 we will prove that our heuristic solution indeed delivers a maximizer for (P ).

57 3.2 The martingale approach 45 First, we consider the problem of maximizing the expected utilitiy of terminal wealth. We check the problem ( S 2 ) max Ψ 2 (Q) = E[U 2 (Q)] ( S 2 ) s.t. E[δ(T )Q] x }{{} =g(q) for the KKT-conditions. We temporarily ignore the restrictions Q, Q F T -measurable, ( ) and treat Q as a simple variable. Moreover, we set C(t) t I. We remark that Ψ 2 is strictly concave, because U 2 is. Ψ 2 C 1 as U 2 C 1. g C 1 and g is convex A solution for ( S 2 ) with positive Lagrangian multiplier λ is also maximizer for our initial problem (S) (without restriction ( ) and the non-negativity of Q and with C(t) ) and for the problem (Ŝ2) { max Ψ2 (Q) (Ŝ2) s.t. E[δ(T )Q] x = The KKT-conditions for ( S 2 ) are 1. g(q) (3.4) Ψ 2 (Q) λ g (Q) = (3.5) λ, λ g(q) = (3.6) We receive from (3.5): Ψ 2 (Q) λ g (Q) = E[U 2(Q) λδ(t )] = U 2(Q) = λδ(t ) > (see theorem A.1) λ > (as δ(t ) > )! I 2 := (U 2) 1 (see theorem A.1) Q = I 2 (λδ(t )) > (see theorem A.1) As we have λ >, we know that g(q) =, because (3.6) must be valid. g(q) = E [δ(t )I 2 (λδ(s))] = x }{{} =:A 2 (λ) A 2 (λ) = x

58 46 Portfolio optimization If we assume that! A 1 2, then we get as the optimal solution (Q ): λ = A 1 2 (A 2(λ)) = A 1 2 (x) > Q ( = I 2 A 1 2 (x)δ(t )) > We have found a heuristic solution for the terminal wealth problem ( S 2 ) (maximization of utility of the terminal wealth as a single problem). Now, we regard the problem ( S 1 ) of maximizing only the utility of consumption which means that our terminal wealth equals zero. ( S 1 ) [ ] max Ψ T 1 (C) = E U1 s (C s )ds [ ] T s.t. E δ(s)c s ds x Suppose now that the solution of ( S 1 ) C is similar to the one of problem ( S 2 ), i.e.: 1. C s = I s 1(λδ(s)) with λ > 2. E δ(s)i1(λδ(s))ds s = x } {{ } =:A 1 (λ) 3.! A 1 1 with λ = A 1 1 (x) ( C t = I1 t A 1 1 (x)δ(t)) >. Remark i) As λ > C solves the problem (Ŝ1) (Ŝ1) [ ] max Ψ T 1 (C) = E U1 s (C s )ds [ ] T s.t. E δ(s)c s ds x =

59 3.2 The martingale approach 47 ii) To see that C is optimal we compute Ψ 1 (C ) = E = E E U1 s (Cs ) ds U s 1 ( ( I s 1 A 1 1 (x)δ(s))) ds ( theorem A.1) ( U s 1 (C s ) + A 1 1 (x)δ(s) (C s C s ) ) ds = Ψ 1 (C) + A 1 1 }{{ (x) } E Cs δ(s)ds E C s δ(s)ds }{{}}{{} =x x Ψ 1 (C) Thus, we solved the problems ( S 2 ) (respectively ( S 1 )) and (Ŝ2) (respectively (Ŝ1)) heuristically. However, strategies leading to a terminal value without consumption or viceversa a consumption without terminal wealth are not realistic and one-sided. The truth lies in between the problems. Consequently, we denote: [ ] max Ψ(Q, T C) = E U ( S) 1 s (C s )ds + U 2 (Q) [ ] T s.t. E δ(s)c s ds + δ(t )Q = x A(λ) = E δ(s)i1(λδ(s))ds s + δ(t )I 2 (λδ(t )) = x A 1 (x) = λ Ct = I1 t ( A 1 (x)δ(t) ) Q ( = I 2 A 1 (x)δ(t ) ) Now, we have to proof on the hand that (Q, C ) is optimal for (P ) and on the other hand that it exists a unique inverse function A 1 as described above. Before we do that we extend our class of utility functions.

60 48 Portfolio optimization 3.4 Definition (Extended utility functions) a) Let u : (, ) R be strictly concave, u C 1 s.t. i) u () := lim x u (x) > and ii) u ( x) = for a unique x (, ] are held. Then u is a utility function. b) Further U : [, T ] (, ) R is a utility function, if it satisfies Remarks i) U t := U(t, ) is utility function for any t I ii) U(, x) C for any fixed x (, ). iii) U t( x) := U x (t, x) = for a unique x (, ] iv) lim x U t(x) > Utility functions in terms of definition 3.4 are utility functions in the sense of definition 3.1 with x = and u () = or rather lim x U t(x) =. Definition 3.4 allows us to deal with a bigger class of utility functions, especially quadratic utility functions (see example 3.5), as often used in finance as criticized for example for the mean-variance-approach. From now on we regard utility functions in the sense of definition Example With α, β, γ R + examples for extended utility functions are: i) u(x) = α β exp( γx) ii) U(t, x) = exp( αt) β(x γ) 2 As we change the definition of utility functions, their properties change as well. 3.6 Lemma (Properties of utility functions) Let u, U be utility functions. Then we have 1. u, U are strictly increasing on (, x] 2. u, U t are strictly decreasing on [ x, ) 3. u C is strictly decreasing for x [, x], u : [, x] [, u ()] 4. U t is strictly decreasing for x [, x] and any fixed t I, U t : {t} [, x] [, U t()] Proof: is similar to the one in appendix A.3.

61 3.2 The martingale approach 49 Figure 3.2: Extended utility function U t (x) = exp( t) 1 2 (x 1 2 )2 3.7 Corollary (Properties of I t 1 and I 2 ) Let be I t 1 and I 2 the unique inverse functions of (U t 1) and U 2. Hence, we get 1. I t 1, I 2 are strictly decreasing on [, (U t 1) ()] or rather [, U 2()] 2. I t 1 C [, (U t 1) ()] 3. I1 t : [, (U1) t ()] [, x 1 ], i.e. lim I t y 1(y) = x 1 and lim I1(y) t = y (U1 t) () 4. I 2 C[, U 2()] 5. I 2 : [, U 2()] [, x 2 ], i.e. lim I 2 (y) = x 2 and lim I y 2(y) = y U 2 () Proof: follows from theorem A.1 and the preceding lemma 3.6. Notation We extend the inverse functions I1 t and I 2 as follows: { I1(y) t I t := 1 (y) if y [, (U1) t ()] else Remarks { I2 (y) if y [, U I 2 (y) := 2()] else As extended versions we see that I t 1 and I 2 C [, ). I1 t and I 2 are strictly decreasing on (, ) under the additional T assumption ξ(s) 2 ds < (compare Korn [2], pp ).